Bol loops and Bruck loops of order pq up to isotopism

Abstract

Let p>q be odd primes. We classify Bol loops and Bruck loops of order pq up to isotopism. When q does not divide p2−1, the only Bol loop (and hence the only Bruck loop) of order pq is the cyclic group of order pq. When q divides p2−1, there are precisely ⌊(p−1+4q)(2q)−1⌋ Bol loops of order pq up to isotopism, including a unique nonassociative Bruck loop of order pq.